1. Web Housing Expert Systems Using New Interval Type-2 Fuzzy Reasoning
Ling Gu
Thesis adviser: Dr. Yanqing Zhang

2. Outline Introduction Fuzzy logic theory System optimizations
Background, Motivation, Contributions
Fuzzy logic theory
Fuzzy sets, Type-1, Type-2, Interval type-2, Upper-lower method
System optimizations
Structure and parameter optimization, Least square method
Web Housing expert
Consideration, Implementation, Examples

3. --- Decision support systems
Background (1)
--- Decision support systems
A decision support system is a program that aids in making decisions and involves queries and constraints that need to be satisfied.
Fuzzy decision support systems, unlike classical logical systems (e.g. expert systems), aim at modeling the imprecise modes of reasoning to make rational decisions in an environment of uncertainty and imprecision [1].

4. --- Fuzzy logic systems
Background (2)
--- Fuzzy logic systems
Fuzzy logic system (FLS) provides a very useful solution for handling vague, ambiguous, and uncertain characteristics of information.
FLS is based on fuzzy set theory which is primarily formulated and developed by Zadeh [4].
Two kinds of fuzzy sets: Type-1 and Type-2 defined by the membership functions are used in FLS.

5. Background (3) --- Related works
Wang et al. used a table-lookup scheme to generate fuzzy rules directly from numerical examples and proved that this fuzzy inference system is a universal approximation [11].
Nozaki et al. presented a heuristic method for generating Takagi-Sugeno fuzzy rules from numerical data [10].
Grauel et al. investigated the connection between the shape of transfer functions and the shape of membership functions [12].
Klawonn applied to construct a fuzzy controller from the training data [13].
Yager proposed the first decision support system using type-2 fuzzy sets [18].
Decision support systems have been applied in various fields including medical diagnosis, business, military, industry, traffic control, and science research.

6. Motivations
Example:
For a dream house, he/she might have various criteria or constraints such that the house is not too far from a highway exit, within 15-minute driving distance from his/her work place, which costs around $300,000, the house is in a good location, the house has a big backyard, and so on. Furthermore, good location has different meaning for different people. Some people would like downtown areas, some people would like a good school district, and some people would like quiet places
To develop a fuzzy decision support system which has the capability of handling multi-criteria and constraints and dealing with vague and imprecise information.

7. Contributions
Applied the interval type-2 fuzzy set into the decision support system to represent the high level uncertainty.
Developed a upper-lower method to handle the complex calculations in the interval type-2 FLS.
Introduced the least square method to optimize the parameters of membership functions to obtain a better performance.

8. Fuzzy Sets
A fuzzy set A of a set X is defined as a set of ordered pairs, each with the first element from X and the second element from an interval of [0, 1]:
A:X  [0, 1].
This defines a mapping, A, between elements of the set X and values in the interval [0, 1]. The value zero is used to represent complete non-membership, the value one is used to represent complete membership, and values in between are used to represent intermediate degrees of membership. The mapping is described by the membership function.

9. Type-1 fuzzy sets
Satisfaction is a linguistic variable which has three linguistic values (bad, fair, and good).
The shape of a membership function can be triangle, Gaussian, or other shapes.

10. Type-2 fuzzy sets Primary membership function
Secondary membership function
Type-2 fuzzy sets are described by membership functions that are themselves fuzzy .

11. Interval type-2 fuzzy sets
Primary membership function
Secondary membership function
For an interval type-2 FLS,  has the same probability (equals to 1) to be any values in its interval range

12. Upper-lower limit method
avg = (H + L ) / 2 ,  = H - avg = avg - L

13. Integrated Fuzzy output
R = Ravg  R

14. Rule base Mamdani-Assilian rule base [23]:
Ri: If x1 is A1, x2 is A2, …, xn is An
then y = Bi .
Used by most fuzzy systems. It is also used in this thesis.
Takagi and Sugeno rule base [24]:
then y = p0 + p1A1 + p2A2 + … + pnAn .
Takagi-Sugeno FLS usually needs a smaller number of rules, because their output is already a linear function of the inputs rather than a constant fuzzy set.

15. Type-1 FLS
Reference 7

16. Type-2 FLS
Reference 19

17. System optimizations
Structure optimization determines the fuzzy partition of input-output variables, and the set of rules to be used to generate mapping between input and output.
By considering the input variables separately
Parameter optimization, which are tuned on the experimental data through optimization procedures, are associated with the membership functions of input-output variables.
The least square method is used to optimize the parameters of the membership functions

18. Least square method
The least square method is the most common technique in data analysis, which assumes that the best-fit curve of a given set of data is the curve that has the minimal sum of the deviations squared [31].
where xi is the independent variable and Yi is the corresponding dependent variable. f(x) is the fitting curve.

19. General Tuning Algorithm
For a FLS with defined membership functions and rules, the general optimization algorithm using the least square will be:
Begin
Read training data
Type-reduce for type-2 membership functions
Select tuning parameters, P1, P2, …, (1, 2, …)*
Initialize tuning parameters and 2
Determine appropriate rules to fire
Calculate out using prod and max operators defined in type-1
Calculate output using the method of the controid of area
Compute squared deviation
Calculate 2(P1, P2, …, 1, 2, …)
Adjust the tuning parameters to find minimum 2 (P1, P2, …, 1, 2, …)
Output tuning results
End
* For type-2 only; for type-1  = 0.

20. Structure of Web Housing Expert
User Interface
Training Process
Fuzzy Logic System
Database System
FLS: interval type-2 fuzzy set
Database: MS access
Training process: least square in C++
User interface: Java, Servlet

21. Database UNITS Price: $1000, Year: Year, and Distance: mile No Address
Location
Year
Distance
S(Exp.) 1
101 1st Ave.
230 7 5 15 2
123 Lakeside Rd.
450 9 30 6 3
204 7th St.
310 8 4
226 Wood St.
400 20
228 Wood St.
350 10
402 North Rd.
150 25
533 3rd Ave.
180
666 Main St.
290
705 Main St.
260
717 South Rd.
200
UNITS
Price: $1000, Year: Year, and Distance: mile

22. Linguistic variables INPUT VARIABLES OUTPUT VARIABLES
Price (0 – 500 k$): Low, Fair, High
Location (0 – 10): Bad, Fair, Good
Year (0 – 10 year): New, Old
Distance (0 – 50 mile): Close, Far
OUTPUT VARIABLES
Satisfaction (0 -10): Bad, Fair, Good

23. Membership Functions (1)
The membership function of price is an interval type-2. It is reduced to type-1 using upper-low limit method. P and  are the adjustable parameters.

24. Membership Functions (2)

25. Reduce Number of Rules 32*22 = 36
The total number of possible rules is
32*22 = 36
Note that
If P is {fair, high} and L is {fair, good} and Y is {new, old} and D is {close, far}
is equivalent to
P is {fair, high} and L is {fair, good}
Because Y and D take every one of the elements from their domains.
The inputs can be divided into two groups (price-location and year-distance) and considered separately, then combine these two groups. The total number of rules becomes
= 22

26. Fuzzy Rules First Group (1) Second Group (2) L P Low Fair High Good
Poor
Bad
D Y
New
Old
Close
Good
Fair
Far
Poor
Combination (C)
1st 2nd
Good
Fair
Poor

27. Parameter Optimization (1)
Read training data
Initialize 2
For  = 0 to 0.2
For p = 100 to 400
For l = 3 to 7
2(, p, l) = 0
For each individual training data
Calculate S using prod and max operators
Update 2(, p, l)
done
If 2(, p, l) smaller than 2
Then 2  2(, p, l)
done (l)
done (p)
done ()
Output , p, l
The product and maximum methods are used in the calculations of S:
e.g. 1 = p*l, 2 = y*d, C = 1*2, and s = max (C1, C2, C3)

28. Parameter Optimization (2)
P changes from 250 to 180
p varies from 0.05 to L shifts from 5 to 5.1

29. Parameter Optimization (3)
House #
S (Exp)
Sbefore (calc)
Safter (calc)
S_Range
Sbefore
Safter 1 7
5.8
7.3
0.547
1.2
0.3 2 6
4.6
5.4
0.482
1.4
0.6 3 8
4.4
7.7
0.368
3.6 4
5.5
6.9
0.258
1.5
0.1 5
8.3
0.3 47
0.7
0.295
1.3
0.2
7.2
6.3
0.496
5.9
7.0
0.331
1.1
0.0 9
6.6
8.1
0.415 10
7.6
6.8
0.346
Sbefore = |S(exp) – Sbefore(calc)|
Safter = |S(exp) – Safter(calc)|

30. User Interface (1)

31. User Interface (2)

32. User Interface (3)

33. User Interface (4)

34. User Interface (5)

35. Procedure 0. Training system.
Obtain user’s queries through the user interface.
Search the database to find matched data.
Determine the fuzzy membership values activated by the inputs through the fuzzy inference system.
Decide which rules are fired in the rule set.
Combine the membership values for each activated rule using the AND operator.
Trace rule activation membership values back through the appropriate output fuzzy membership functions.
Utilize defuzzification to calculate the value for the output variable.
Rank decisions according to the output values using database system.
Display the output values on the user interface.

36. Type-2 vs. Type-1
Type-2 FLS: precise and reasonable

37. Interval Output
Type-2 offers an interval satisfaction instead of the crisp output from type-1. The interval also gives more freedom to describe search results for a same query from different people because people will have different answer to a same question.
Some people might think the price of house 1 is too high and some people might think the price of house 1 is reasonable based on their own view about the house.
Dell Inspiron 9300 is a very good laptop to programmers but it may be too heavy (8.2 pounds) for a traveler or too expansive ($2,858) for a student.

38. A Real Case (1) * Computer Shopper (issues July – October, 2005) No.
Company
Model
Price
($)
CPU
(GHz)
Memory
(MB)
Display
(inch)
Rank* 1
ABS
Mayhem G4
2199
2.1
1024
15.4
6.2 2
Acer Aspire
As2003Lmi
1599
1.6
512
5.5 3
Acer
Aspire 5000
1099
6.8 4
Averatec
4200
1199
5.6 5
Dell
Inspiron 9300
2842 17
7.8 6
Gateway
M460S
1359
1.86 15
6.4
* Computer Shopper (issues July – October, 2005)

39. A Real Case (2)

40. A Real Case (3)

41. Conclusions
An interval type-2 FLS based decision support system, the Web Housing Expert for online users is proposed.
A fuzzy output of the interval type-2 FLS based decision support system can be obtained using the upper-lower limit technique.
The total number of rules is reduced from 36 to 22 by separately considering the input variables. The membership functions of price and location are optimized using the least square technique to achieve the best performance.
It shows that the interval type-2 FLS not only provides a more reasonable result but also a more accurate result than type-1 FLS does

42. Future Work
Introduce more input variables to describe houses more precisely
Use more type-2 variables to handle the uncertainties of words effectively
Weight different input variables to present the importance of different variables
Apply interval inputs to solve the problem of the precise search criteria

43. ACKNOWLEDGEMENTS
I would like to send special thanks to my thesis advisor Dr. Zhang for his help, guidance and support throughout the duration of the thesis.
I also would like to thank my thesis committee members, Dr. Sunderraman and Dr. Zhu.